Topics in Algebraic and Topological K-Theory


Author: Paul Frank Baum,Ralf Meyer,Rubén Sánchez-García,Marco Schlichting,Bertrand Toën
Publisher: Springer Science & Business Media
ISBN: 3642157076
Category: Mathematics
Page: 308
View: 3743

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This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.

Space – Time – Matter

Analytic and Geometric Structures
Author: Jochen Brüning,Matthias Staudacher
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110452154
Category: Mathematics
Page: 517
View: 2881

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This monograph describes some of the most interesting results obtained by the mathematicians and physicists collaborating in the CRC 647 "Space – Time – Matter", in the years 2005 - 2016. The work presented concerns the mathematical and physical foundations of string and quantum field theory as well as cosmology. Important topics are the spaces and metrics modelling the geometry of matter, and the evolution of these geometries. The partial differential equations governing such structures and their singularities, special solutions and stability properties are discussed in detail. Contents Introduction Algebraic K-theory, assembly maps, controlled algebra, and trace methods Lorentzian manifolds with special holonomy – Constructions and global properties Contributions to the spectral geometry of locally homogeneous spaces On conformally covariant differential operators and spectral theory of the holographic Laplacian Moduli and deformations Vector bundles in algebraic geometry and mathematical physics Dyson–Schwinger equations: Fix-point equations for quantum fields Hidden structure in the form factors ofN = 4 SYM On regulating the AdS superstring Constraints on CFT observables from the bootstrap program Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities Yangian symmetry in maximally supersymmetric Yang-Mills theory Wave and Dirac equations on manifolds Geometric analysis on singular spaces Singularities and long-time behavior in nonlinear evolution equations and general relativity

2016 MATRIX Annals


Author: Jan de Gier,Cheryl E. Praeger,Terence Tao
Publisher: Springer
ISBN: 3319722999
Category: Mathematics
Page: 656
View: 2324

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MATRIX is Australia’s international, residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each lasting 1-4 weeks. This book is a scientific record of the five programs held at MATRIX in its first year, 2016: Higher Structures in Geometry and Physics (Chapters 1-5 and 18-21); Winter of Disconnectedness (Chapter 6 and 22-26); Approximation and Optimisation (Chapters 7-8); Refining C*-Algebraic Invariants for Dynamics using KK-theory (Chapters 9-13); Interactions between Topological Recursion, Modularity, Quantum Invariants and Low-dimensional Topology (Chapters 14-17 and 27). The MATRIX Scientific Committee selected these programs based on their scientific excellence and the participation rate of high-profile international participants. Each program included ample unstructured time to encourage collaborative research; some of the longer programs also included an embedded conference or lecture series. The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on selected topics related to the MATRIX program; the remaining contributions are predominantly lecture notes based on talks or activities at MATRIX.

Lecture Notes in Algebraic Topology


Author: James Frederic Davis,Paul Kirk
Publisher: American Mathematical Soc.
ISBN: 0821821601
Category: Mathematics
Page: 367
View: 2575

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The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems. To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated. Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book. The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic $K$-theory and the s-cobordism theorem. A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the ``big picture'', teaches them how to give mathematical lectures, and prepares them for participating in research seminars. The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.

Algebraic K-Theory


Author: Vasudevan Srinivas
Publisher: Springer Science & Business Media
ISBN: 0817647368
Category: Mathematics
Page: 341
View: 477

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Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and students in mathematics. This book is based on lectures given by the author at the Tata Institute in Bombay and elsewhere. This new edition includes an appendix on algebraic geometry that contains required definitions and results needed to understand the core of the book.

Handbook of K-Theory


Author: Eric Friedlander,Daniel R. Grayson
Publisher: Springer Science & Business Media
ISBN: 354023019X
Category: Mathematics
Page: 626
View: 9246

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This handbook offers a compilation of techniques and results in K-theory. Each chapter is dedicated to a specific topic and is written by a leading expert. Many chapters present historical background; some present previously unpublished results, whereas some present the first expository account of a topic; many discuss future directions as well as open problems. It offers an exposition of our current state of knowledge as well as an implicit blueprint for future research.

Algebraic K-Theory


Author: Richard G. Swan
Publisher: Springer
ISBN: 3540359176
Category: Mathematics
Page: 264
View: 6325

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From the Introduction: "These notes are taken from a course on algebraic K-theory [given] at the University of Chicago in 1967. They also include some material from an earlier course on abelian categories, elaborating certain parts of Gabriel's thesis. The results on K-theory are mostly of a very general nature."

Cohomology of Groups and Algebraic K-theory


Author: Lizhen Ji,Kefeng Liu,Shing-Tung Yau
Publisher: International Press of Boston
ISBN: 9781571461445
Category: Mathematics
Page: 517
View: 9470

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ALM Published jointly by International Press and by Higher Education Press of China, the Advanced Lectures in Mathematics (ALM) series brings the latest mathematical developments worldwide to both researchers and students. Each volume consists of either an expository monograph or a collection of significant introductions to important topics. The ALM series emphasizes discussion of the history and significance of each topic discussed, with an overview of the current status of research, and presentation of the newest cutting-edge results. Cohomology of Groups and Algebraic K-theory Cohomology of groups is a fundamental tool in many subjects of modern mathematics. One important generalized cohomology theory is the algebraic K-theory. Indeed, algebraic K-groups of rings are important invariants of the rings and have played important roles in algebra, topology, number theory, etc. This volume consists of expanded lecture notes from a 2007 seminar at Zhejiang University in China, at which several leading experts presented introductions, to and surveys of, many aspects of cohomology of groups and algebraic K-theory, along with their broad applications. Two foundational papers on algebraic K-theory by Daniel Quillen are also included.

Topics in K-Theory

The Equivariant Künneth Theorem in K-Theory. Dyer-Lashof operations in K-Theory
Author: L.H. Hodgkin,V.P. Snaith
Publisher: Springer
ISBN: 3540380264
Category: Mathematics
Page: 294
View: 3945

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Michael Atiyah Collected Works

Volume 2: K-Theory
Author: Michael Atiyah
Publisher: Oxford University Press
ISBN: 9780198532767
Category: Language Arts & Disciplines
Page: 854
View: 8197

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One of the greatest mathematicians in the world, Michael Atiyah has earned numerous honors, including a Fields Medal, the mathematical equivalent of the Nobel Prize. While the focus of his work has been in the areas of algebraic geometry and topology, he has also participated in research with theoretical physicists. For the first time, these volumes bring together Atiyah's collected papers--both monographs and collaborative works-- including those dealing with mathematical education and current topics of research such as K-theory and gauge theory. The volumes are organized thematically. They will be of great interest to research mathematicians, theoretical physicists, and graduate students in these areas.

Algebraic K-theory of Crystallographic Groups

The Three-Dimensional Splitting Case
Author: Daniel Scott Farley,Ivonne Johanna Ortiz
Publisher: Springer
ISBN: 3319081535
Category: Mathematics
Page: 148
View: 3587

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The Farrell-Jones isomorphism conjecture in algebraic K-theory offers a description of the algebraic K-theory of a group using a generalized homology theory. In cases where the conjecture is known to be a theorem, it gives a powerful method for computing the lower algebraic K-theory of a group. This book contains a computation of the lower algebraic K-theory of the split three-dimensional crystallographic groups, a geometrically important class of three-dimensional crystallographic group, representing a third of the total number. The book leads the reader through all aspects of the calculation. The first chapters describe the split crystallographic groups and their classifying spaces. Later chapters assemble the techniques that are needed to apply the isomorphism theorem. The result is a useful starting point for researchers who are interested in the computational side of the Farrell-Jones isomorphism conjecture, and a contribution to the growing literature in the field.

Topics in Cohomological Studies of Algebraic Varieties

Impanga Lecture Notes
Author: Piotr Pragacz
Publisher: Springer Science & Business Media
ISBN: 3764373423
Category: Mathematics
Page: 300
View: 5749

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The articles in this volume study various cohomological aspects of algebraic varieties: - characteristic classes of singular varieties; - geometry of flag varieties; - cohomological computations for homogeneous spaces; - K-theory of algebraic varieties; - quantum cohomology and Gromov-Witten theory. The main purpose is to give comprehensive introductions to the above topics through a series of "friendly" texts starting from a very elementary level and ending with the discussion of current research. In the articles, the reader will find classical results and methods as well as new ones. Numerous examples will help to understand the mysteries of the cohomological theories presented. The book will be a useful guide to research in the above-mentioned areas. It is adressed to researchers and graduate students in algebraic geometry, algebraic topology, and singularity theory, as well as to mathematicians interested in homogeneous varieties and symmetric functions. Most of the material exposed in the volume has not appeared in books before. Contributors: Paolo Aluffi Michel Brion Anders Skovsted Buch Haibao Duan Ali Ulas Ozgur Kisisel Piotr Pragacz Jörg Schürmann Marek Szyjewski Harry Tamvakis

An Introduction to K-Theory for C*-Algebras


Author: M. Rørdam,Flemming Larsen,N. Laustsen
Publisher: Cambridge University Press
ISBN: 9780521789448
Category: Mathematics
Page: 242
View: 9364

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This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.

Algebraic and Geometric Topology

Proceedings of a Conference held at Rutgers University, New Brunswick, USA, July 6-13, 1983
Author: Andrew Ranicki,Norman Levitt,Frank Quinn
Publisher: Springer
ISBN: 3540394133
Category: Mathematics
Page: 426
View: 4936

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Algebraic K-theory and Algebraic Number Theory

Proceedings of a Seminar Held January 12-16, 1987, with Support from the National Science Foundation and Japan Society for the Promotion of Science
Author: Michael R. Stein,R. Keith Dennis
Publisher: American Mathematical Soc.
ISBN: 0821850903
Category: Mathematics
Page: 488
View: 7960

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Algebraic Topology


Author: C. R. F. Maunder
Publisher: Courier Corporation
ISBN: 9780486691312
Category: Mathematics
Page: 375
View: 2975

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Based on lectures to advanced undergraduate and first-year graduate students, this is a thorough, sophisticated, and modern treatment of elementary algebraic topology, essentially from a homotopy theoretic viewpoint. Author C.R.F. Maunder provides examples and exercises; and notes and references at the end of each chapter trace the historical development of the subject.